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Optimizing fermionic Hamiltonians with classical interactions

Maarten Stroeks, Barbara M. Terhal, Yaroslav Herasymenko

TL;DR

This work establishes rigorous constant-factor guarantees for approximating the ground energy of fermionic Hamiltonians with classical (diagonal) interactions using fermionic Gaussian states. By introducing a Gaussian blend, it constructs Gaussian covariances that blend diagonal and hopping components and proves a universal $1/3$-approximation for traceless CIFH optimization, independent of sparsity. It further provides efficient, constructive schemes in structured settings: a polynomial-time bipartite-graph method for the classically interacting part, a Goemans–Williamson–style SDP achieving $0.637$ for PSD CIFH optimization, and a Gaussian-state-based approach achieving $1/2$ for Fermionic Max Cut, with extensions to fixed particle number. The results offer a principled basis for Hartree–Fock-type methods in quantum chemistry and condensed-matter physics, and introduce the Gaussian blend as a versatile toolkit for energy approximation in fermionic systems.

Abstract

We consider the optimization problem (ground energy search) for fermionic Hamiltonians with classical interactions. This QMA-hard problem is motivated by the Coulomb electron-electron interaction being diagonal in the position basis, a fundamental fact that underpins electronic-structure Hamiltonians in quantum chemistry and condensed matter. We prove that fermionic Gaussian states achieve an approximation ratio of at least 1/3 for such Hamiltonians, independent of sparsity. This shows that classical interactions are sufficient to prevent the vanishing Gaussian approximation ratio observed in SYK-type models. We also give efficient semi-definite programming algorithms for Gaussian approximations to several families of traceless and positive-semidefinite classically interacting Hamiltonians, with the ability to enforce a fixed particle number. The technical core of our results is the concept of a Gaussian blend, a construction for Gaussian states via mixtures of covariance matrices.

Optimizing fermionic Hamiltonians with classical interactions

TL;DR

This work establishes rigorous constant-factor guarantees for approximating the ground energy of fermionic Hamiltonians with classical (diagonal) interactions using fermionic Gaussian states. By introducing a Gaussian blend, it constructs Gaussian covariances that blend diagonal and hopping components and proves a universal -approximation for traceless CIFH optimization, independent of sparsity. It further provides efficient, constructive schemes in structured settings: a polynomial-time bipartite-graph method for the classically interacting part, a Goemans–Williamson–style SDP achieving for PSD CIFH optimization, and a Gaussian-state-based approach achieving for Fermionic Max Cut, with extensions to fixed particle number. The results offer a principled basis for Hartree–Fock-type methods in quantum chemistry and condensed-matter physics, and introduce the Gaussian blend as a versatile toolkit for energy approximation in fermionic systems.

Abstract

We consider the optimization problem (ground energy search) for fermionic Hamiltonians with classical interactions. This QMA-hard problem is motivated by the Coulomb electron-electron interaction being diagonal in the position basis, a fundamental fact that underpins electronic-structure Hamiltonians in quantum chemistry and condensed matter. We prove that fermionic Gaussian states achieve an approximation ratio of at least 1/3 for such Hamiltonians, independent of sparsity. This shows that classical interactions are sufficient to prevent the vanishing Gaussian approximation ratio observed in SYK-type models. We also give efficient semi-definite programming algorithms for Gaussian approximations to several families of traceless and positive-semidefinite classically interacting Hamiltonians, with the ability to enforce a fixed particle number. The technical core of our results is the concept of a Gaussian blend, a construction for Gaussian states via mixtures of covariance matrices.

Paper Structure

This paper contains 21 sections, 20 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Fermionic Gaussian states
  5. Covariance matrix
  6. Optimization over Gaussian states
  7. Optimization of quadratic Hamiltonians as a semi-definite program
  8. Blending Gaussian states
  9. Proof of Theorem \ref{['theorem:maintraceless']}: existence of constant-ratio Gaussian approximations.
  10. Optimization procedure
  11. Efficient approximate constructions
  12. Proof of Theorem \ref{['theorem:bipartitetraceless']}: classical interactions on bipartite graphs
  13. Proof of Theorem \ref{['theorem:maintraceful']}: positive semi-definite classical interactions
  14. Fermionic Max Cut
  15. Fermionic optimization in the presence of a particle constraint
  16. ...and 6 more sections

Key Result

Theorem 2.2

There exists a pure fermionic Gaussian state $\rho$ that achieves an approximation ratio $\frac{1}{3}$ for Traceless CIFH Optimization (Problem prob:maintracelessoptimizationproblem).

Figures (1)

  • Figure 1: Approximation ratio $r$ versus $p_{\mathrm{class}}$ for $3$-site (i.e., $6$-mode) traceless Fermi-Hubbard Hamiltonian. The maximum approximation ratio $r_{*}\:(>1/3)$ is achieved at an intermediate value of $p_{\rm class}$.

Theorems & Definitions (43)

  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.5
  • Theorem 2.7
  • Definition 3.1: Fermionic Gaussian states
  • Remark 3.2
  • Definition 3.4: Slater determinant and classical states
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 33 more